The Expectations Hypothesis of the Term Structure: Some Empirical Evidence for Portugal

Munich Personal RePEc Archive

The Expectations Hypothesis of the Term Structure: Some Empirical Evidence for Portugal

Silva Lopes, Artur C. and Monteiro, Olga Susana

CEMAPRE ISEG- UTL, Banco de Portugal

2007

Online at https://mpra.ub.uni-muenchen.de/6310/

MPRA Paper No. 6310, posted 15 Dec 2007 06:33 UTC

The Expectations Hypothesis of the Term

Structure: Some Empirical Evidence for Portugal

Artur C. B. da Silva Lopes^† CEMAPRE and ISEG, Technical University of Lisbon Olga Susana M. Monteiro

Departamento de Estat´ıstica Banco de Portugal December 14, 2007

Abstract

The purpose of this paper is to test the (rational) expectations hypothesis of the term structure of interest rates using Portuguese data for the interbank money market. The results obtained support only a very weak, long-run or “as- ymptotic” version of the hypothesis, and broadly agree with previous evidence for other countries.

The empirical evidence supports the cointegration of Portuguese rates and the “puzzle” well known in the literature: although its forecasts of future short- term rates are in the correct direction, the spread between longer and shorter rates fails to forecast future longer rates. In the single equation framework, the implications of the hypothesis in terms of the predictive ability of the spread are also clearly rejected, even for the more stable period which emerged in the middle nineties.

Keywords: term structure of interest rates; expectations hypothesis; hypothe- sis testing; cointegration; Portugal.

JEL classification: E43, C22, C32.

∗The first author gratefully acknowledges financial support from Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT) through Programa POCTI, partially funded by FEDER.

†Corresponding author. E-mail: asl(at)iseg.utl.pt. Address: ISEG, gab. 315, Rua do Quelhas 6, 1200 781 Lisboa, Portugal.

1 Introduction

The expectations hypothesis (EH) of the term structure of interest rates, which states that the observed term structure can be used to infer market participants’

expectations about future interest rates, has been at the origin of an extraordinary amount of econometric analysis; see, e.g., Campbell (1995), Campbell and Shiller (1987, 1991), Engsted and Tanggaard (1994a,b), Hall et al. (1992), Hardouvelis (1994), Jondeau and Ricart (1999), Lanne (2000), Sarno et al. (2005), Thornton (2006), and Tzavalis (2003).

Understanding the term structure of interest rates has always been viewed as crucial to assess the impact of monetary policy and its transmission mechanism, to predict interest rates, exchange rates and economic activity, and to provide infor- mation about expectations of participants in financial markets. In this paper, the EH of the term structure of interest rates, embedding the rational expectations hy- pothesis, is tested with Portuguese data for interbank money market (IMM) rates.

To the best of our knowledge, this paper contains the first examination of the EH using Portuguese IMM data.

Some of the alternative ways of testing the hypothesis are briefly reviewed, both in the framework of single and multiple equation models. We focus particularly on cointegration analysis and on the predictive ability of the spread.

The results obtained support only a very weak version of the hypothesis and are in line with most of the conclusions in the literature. The empirical evidence supports the cointegration hypothesis of Portuguese rates and the “puzzle” well known in the literature of the EH: although forecasts of short-term rates changes based on the spread are in the correct direction, it fails in forecasting future longer rates because the forecasts are in the wrong direction. More importantly, in the single equation framework, the strict implications of the hypothesis on the predictive ability of the spread are clearly rejected by our data. Hence, our evidence closely agrees with most of the previous results in the literature; see, inter alia, Arshanapalli and Doukas (1994), Campbell and Shiller (1991), Evans and Lewis (1994), Hurn et al.

(1995), Jondeau and Ricart (1999), Thornton (2006), Tzavalis (2003) and Tzavalis and Wickens (1997).

To summarize all the evidence we propose that such a weak version of the hy- pothesis is called long-run or asymptotic, a term which we borrow from the rational

expectations hypothesis literature. Moreover, contrasting with most empirical stud- ies, we provide an historically and statistically based sample-split analysis, which confirms and reinforces the results for the whole sample.

Actually, this quest for robustness is also a trait of our study. Instead of relying on a single model/method, we diverge from previous empirical assessments of the EH in the range of methods and models that we use. For instance, although relatively standard, we employ two techniques rarely (if ever) used to test the EH: DOLS estimation and testing and t-ecm tests for cointegration. As another example, we employ several VAR and VECM models chosen with different criteria. Naturally, this allows us to robustify our inferences.

The remainder of this paper is organized as follows. Some of the most important implications and testing procedures of the EH are reviewed in the next section. In section 3 we describe the data that we have used and in section 4 we present the main empirical evidence. In section 5 we discuss the sample-split point and reassess the evidence considering a partition of the sample. Section 6 concludes the paper.

2 Some implications and testing procedures of the EH

2.1 In single equation models

In the single equation setup the focus is on pairs of interest rates. Some of the available tests regarding the spread between interest rates of different maturities are described below.

2.1.1 Cointegration

Since nominal interest rates are bounded below by zero, the I(1) property cannot be strictly justified on theoretical grounds. However, their typical high persistent behaviour in response to shocks has led to an almost universal consensus about the presence of a unit root. Hence, cointegration methods are applicable.

Assuming that interest rates correspond to I(1) processes, the EH requires coin- tegration between interest rates with different maturities. Denoting the long and the short rates with r_t⁽ⁿ⁾ and r^(m)_t , respectively, the stationarity of the spread, S_t^(n,m)=r⁽ⁿ⁾_t −r^(m)_t , is a necessary, although not sufficient, condition for the EH to hold, as it is an implication of several term structure models. In fact, as is sometimes

pointed out, more traditional theories also demand this condition; see, e.g., Lanne (2000), Patterson (2000), and Taylor (1992).

If the spread is stationary, then the term/risk premium is also stationary and interest rates are driven by a common stochastic trend, preventing them from drifting too far apart from the equilibrium, so that profitable arbitrage opportunities do not persist. The rate of inflation is the most obvious candidate to represent this common trend (Dom´ınguez and Novales, 2000, Engsted and Tanggaard, 1994b).

2.1.2 The spread as a predictor of interest rate changes

The fundamental equation characterizing the EH states that the long-term interest rate equals an average of current and expected short-term interest rates over the life of the long-term interest rate plus a constant term, representing the time invariant term/risk premium (Φ⁽ⁿ⁾):

r_t⁽ⁿ⁾= 1 k

k−1X

i=0

E_t[r_t+im^(m) ] +Φ⁽ⁿ⁾, (1) wherekis an integer denotingn/m. Expectations formulated at timetfor the future evolution of short-term interest rates drive the longer-term interest rate. When short-term interest rates are expected to rise, longer-term interest rates will also rise.

Using equation (1) it is straightforward to get S_t^(n,m)=Et

hS^∗_t^(n,m)i+Φ⁽ⁿ⁾ =

kX−1 i=1

k−i k Et

h∆r_t+im^{(m) i}+Φ⁽ⁿ⁾, (2)

where S_t^∗(n,m) denotes the perfect foresight spread and∆r_t+im^(m) =r_t+im^(m) −r_t+(i^(m)_−1)m. Hence, the spread is a weighted average (with declining weights) of expected changes of short-term interest rates plus the term/risk premium. Since the spread is such an optimal predictor, a test for the validity of the EH may be based on the equation

S_t^∗(n,m)=δ₀+δ₁S_t^(n,m)+ξ_t, (3) where δ₀ represents −Φ⁽ⁿ⁾, testing H₀ :δ₁ = 1 vs H₁ :δ₁ 6= 1. It should be noted that the error term of this equation is a MA(n−m) process; see, e.g., Evans and Lewis (1994), Gerlash and Smets (1997) and Thornton (2005, 2006) for a closer look at this test.

Continuing to focus on the long-term behaviour of short-term rates, no other variable besides the spread should provide any help for predicting short-term interest rates changes. Therefore, in equation

S_t^∗(n,m)=δ₀+δ₁S_t^(n,m)+δ₂⁰ x_t+η_t, (4) where x_t denotes a vector of variables other than the spread, the EH demands that δ₁= 1 and δ₂ = 0.

Changing the focus to the short-term behaviour of long-term interest rates, an- other important characterization of the EH is provided by

Et

hr_t+m^(n−m)i−r⁽ⁿ⁾_t = m

n−m(S_t^(n,m)−φ^(n,m)_h ), (5) i.e., the expected (short-term) change of the long-term interest rate is defined as a proportion of the difference between the spread and the holding period term premium (φ^(n,m)_h ). When the long-term interest rate is expected to rise over the nextmperiods (in the short-term), potential capital losses are predictable. Therefore, the current long-term interest rate has to be higher than the short-term rate.

If the EH is true, the spread is also the optimal predictor of (short-term) changes of long-term interest rates. Based on equation (5), another EH test can be specified.

As in equation (3), the simpler version tests whether λ₁ = 1 in r_t+m^(n−m)−r_t⁽ⁿ⁾=λ₀+λ₁

· m

n−mS_t^(n,m)

¸

+u_t+m, (6)

and the augmented version is similar to the one of equation (4).

Concerning the predictive ability of the spread, the available empirical evidence tends to agree that:

a) the spread predicts the (long-term) changes in the short-term rates in the direc- tion stated by the EH (ˆδ₁ is generally positive, although sometimes statistically different from unity);

b) however, the spread does not predict the (short-term) changes in long-term rates in the direction required by the EH (usually ˆλ₁ is negative and signifi- cantly distinct from unity).

This is the “puzzle” well known in the EH literature, also known as the “Campbell- Shiller paradox”. Besides providing a recent survey on previous attempts to solve

this puzzle, Thornton (2006) demonstrates that it can emerge very often when the EH does not hold.

2.2 In multiple equation models

In the multiple equation model framework the EH has two cointegration implica- tions:

i) in a system of l interest rates with different maturities there should be one (and only one) common stochastic trend, which is responsible for the long-run movement of all interest rates, and

ii) in each of thel−1 cointegrating vectors the coefficients should sum zero.

While i) should be clear from the previous subsection, the restrictions of ii) de- serve a closer look. Consideringm= 1 and computing equation (1) for all maturities τi,i= 2, ..., l:

r_t^(τi)= 1 τi

τXi−1 i=0

E_t[r⁽¹⁾_t+i] +Φ^(τi)= 1 τi

τXi−1 i=1

E_t[r⁽¹⁾_t+i] + 1 τi

r_t⁽¹⁾+Φ^(τi). But since _τ¹

i

Pτ_i−1

i=1 Et[−r_t⁽¹⁾] =−r_t⁽¹⁾+_τ¹

ir_t⁽¹⁾, the previous equation may be written as

r^(τ_t ⁱ⁾= 1 τ_i

τXi−1 i=1

E_t[r_t+i⁽¹⁾ −r_t⁽¹⁾] +r⁽¹⁾_t +Φ^(τi).

Taking a linear combination of all interest rates in the system,β₁r_t⁽¹⁾+β₂r_t^(τ2)+...+ β_lr^(τ_t ^l) and using the previous equation forr^(τ_t ⁱ⁾, we get, apart from a constant term:

(β₁+β₂+...+β_l)r⁽¹⁾_t +β₂ τ₂

τX2−1 i=1

E_t^hr⁽¹⁾_t+i−r_t⁽¹⁾ⁱ+...+β_l τ_l

τXl−1 i=1

E_t^hr_t+i⁽¹⁾ −r_t⁽¹⁾ⁱ. (7) Now, if interest rates correspond to I(1) processes the spreads will be I(0). Hence, the process of equation (7) will be I(0) iffβ1+β2+...+β_l= 0.

Both implications can be tested in the context of a VAR model using the popular Johansen’s approach (Johansen, 1995). Clearly, this is a case where the inclusion of a deterministic trend appears highly unreasonable. However, a constant term is required. But then, should a restricted or an unrestricted intercept be considered?

While allowing for an unrestricted intercept appears implausible, there is a statistical justification for doing it: ”in vector error-correction models the cointegration rank test based on the unconstrained estimator has somewhat better local power than the test based on the constrained estimator” (Lanne, 2000).

For the cointegration rank analysis we have used trace test statistics. The zero- sum restrictions are tested employing likelihood ratio statistics. Besides these tests, Johansen’s methodology also provides a test for the predictive ability of the spread concerning short-term interest rate changes. In order to do this, one must focus on the factor loadings (usually denoted with α_ij), which measure the influence of the error correction term in each equation. Under the EH, these coefficients should be statistically significant in all equations except in the one for the longer-term interest rate. In other words, the longer-term rate should be weakly exogenous for the cointegration vectors. Moreover, assuming that the vector of interest rates is ordered in ascending order of maturity, the spread will predict in the direction indicated by the EH if the α_ij coefficients are negative for i 6=l and j= 1, ..., l−1.

3 The data

The most natural representation of the term structure of interest rates is with spot rates. But as zero coupon bonds are typically issued for maturities less than a year (short end of the maturity spectrum), spot rates have to be estimated from coupon bonds data for longer maturities. For long periods of time, this work has already been done for some countries but not for Portugal. Since this estimation is beyond our present purposes, a preliminary step of identifying alternative datasets was taken.

This allowed us to get data for a 10-year government bond yield. Although we have used also this dataset at an initial stage, the rather limited scope of the results (available from the authors) lead us to omit their presentation.

For the short end of the term structure, Treasury bills data are the most common alternative. However, for the Portuguese case the number of missing observations is extremely high. At the end, interbank money market (IMM) rates were selected for several reasons. First, they represent the alternative providing the largest number of observations. Second, IMMs tend to be highly competitive, well integrated with other money markets, and internationally comparable. Finally, contrarily to the bond market, the IMM is much less influenced by large institutions aiming portfolio

immunisation.

Monthly data for IMM rates for 1, 3 and 6 months – “value date of same day” – are available at the website of Banco de Portugal (section B.10). Our dataset covers the period from January 1989 to April 2004, i.e., T = 184. For the missing observations (2, 17 and 40 for r_t⁽¹⁾, r⁽³⁾_t and r⁽⁶⁾_t , respectively) some alternatives were considered. Firstly, several univariate and multivariate models were applied to the first differences of the interest rates. However, we have not found the results satisfactory, particularly at the end of the sample. Moreover, when using multivariate models it was not possible to estimate all missing observations.

Hence, using observations from the other two segments of the IMM appeared as an attractive and simple alternative¹. Data for the three segments of the IMM are very highly correlated and it is common to observe that when there are no transactions in one of them the remaining two present some recorded operations. However, this estimation procedure did not allow us obtaining all the missing observations.

Therefore we decided to adopt a two step procedure:

i) in the first stage, whenever possible, missing data were estimated with the monthly variation for rates with the same maturity but “value date deferred 1 or 2 days”;

ii) for the remaining missing observations (20 for r⁽⁶⁾_t only), several alternative models were considered and a simple multivariate model in first differences, relating∆r_t⁽⁶⁾ with∆r⁽¹⁾_t and ∆r_t⁽³⁾ and minimizing MSFE was chosen.

4 Empirical results

4.1 In single equation models

Due to space constraints, some of the results for the single equation approach are only briefly presented. However, all the results are available from the authors. First, preliminary unit root testing, using ADF, PP (Phillips-Perron) and WS (weighted symmetric, see Pantula et al., 1994) tests with several lag truncation parameters

1These two other segments are reported as “value date deferred one or two days”, that is, the transactions are contracted in one day but the transference of funds occurs only one or two days later.

(k), provide overwhelming confirmation evidence for the I(1) hypothesis of interest rates.

Second, the same unit root tests, which may now be viewed as restricted coin- tegration tests, strongly support the stationarity of the spreads, i.e., cointegration with unit cointegration parameters. Augmented Engle-Granger tests (see table 1) provide somewhat weaker evidence for cointegration but this appears to result only from the usual poor power behaviour of these tests. These were performed with fixed (k= 6 and 12) and estimated lag truncation parameters, using the general-to- specific t-sig procedure, denoted with GSt-sig, and the AIC+2 rule (denoted with AIC+2), as recommended by Pantula et al. (1994), using k_MAX= 18.

Table 1. P-values for augmented Engle-Granger tests for cointegration

dependent variable: r_t⁽ⁿ⁾ dependent variable: r^(m)_t rates k= 6 k= 12 GS t-sig AIC+2 k= 6 k= 12 GSt-sig AIC+2 r_t⁽³⁾,r⁽¹⁾_t 0.192 0.379 0.033 0.047 0.172 0.360 0.040 0.060 r_t⁽⁶⁾,r⁽³⁾_t 0.223 0.177 0.083 0.128 0.215 0.214 0.100 0.168 r_t⁽⁶⁾,r⁽¹⁾_t 0.202 0.277 0.035 0.178 0.181 0.298 0.119 0.119

On the other hand, assuming weak exogeneity (see below, table 7), table 2 re- ports the orders, (r, s), of the estimated bivariate ADL models, chosen following the GS t-sig strategy and starting with rMAX = sMAX = 12, together with the t-ecm test statistics for cointegration. As the small sample 5% critical value is −3.232 (see Ericsson and MacKinnon, 2002), these provide very strong evidence for cointegra- tion when the dependent variables are the shorter-term rates. As the homogeneity restriction was not imposed, this favourable evidence must be viewed with some caution. However, DOLS estimation and testing (see table 8 below) provide clear evidence for unit cointegration parameters.

Table 2. t-ecm test statistics for cointegration

dependent variable: r_t⁽ⁿ⁾ dependent variable: r^(m)_t

rates ADL t-ecm ADL t-ecm

r⁽³⁾_t ,r_t⁽¹⁾ ADL(10,3) −2.086 ADL(8,9) −5.431 r⁽⁶⁾_t ,r_t⁽³⁾ ADL(7,5) −4.757 ADL(6,7) −11.15 r⁽⁶⁾_t ,r_t⁽¹⁾ ADL(10,4) −2.509 ADL(4,7) −5.313

Hence, in general terms, the analysis of the long-run properties of the data is strongly favourable to the EH. A much different picture is observed when more de- manding implications are examined. Table 3 contains the results concerning equa- tion (3), evaluating the predictive ability of the spread for short rate changes ². Although the sign of the estimates agrees with the EH, i.e., the predictions are in the correct direction, the restrictions it implies are very clearly and strongly rejected.

Despite this evidence, the spread contains useful information about the future (long- run) behaviour of short-term interest rates, that is,δ^b₁ is significant in all equations.

Table 3. The spread as a predictor of short rate changes (equation (3))

spread bδ₀ bδ₁ EH p-val.

S_t^(3,1) −0.077 0.238 0.000 S_t^(6,3) −0.126 0.423 0.000 S_t^(6,1) −0.235 0.491 0.000

Note: for the calculation of the Wald test statistics we have used a Newey-West correction with a Bartlett kernel and a bandwidth of n−m, but similar results arise when a (fixed) bandwidth equal to 12 is employed.

Then, as expected, the EH is still strongly rejected when ∆r⁽ⁿ⁾_t , representing the short-run dynamics of the longer-term interest rate, is added as an additional regressor to equation (3) (cf. equation 4). However, the spread retains its statistical significance in all the regressions.

Turning to the predictive ability of the spread in respect to longer rate changes, we could not find a single trace of evidence for the validity of the EH. Table 4 contains the results for equation (6): all the estimates are in the incorrect predictive direction and all thep-values for the restrictions implied by the EH are equal to zero.

Moreover, the spread does not seem to contain any relevant information about the future (short-run) behaviour of longer-term interest rates. Obviously, when ∆r^(m)_t , which represents the short-run dynamics of the short-term interest rate, is included as an additional regressor, the evidence against the EH is confirmed.

To sum up, in single equation models the empirical evidence is mixed: on one hand, the long-run properties of the data are clearly supportive of the hypothesis;

2Previously, Wu-Hausman exogeneity tests were performed, providing no evidence for the in- consistency of the OLS estimator. A similar preliminary analysis was performed also in relation to equation (6), providing the same type of results.

Table 4. The spread as a predictor of long rate changes (equation (6))

spread bλ0 bλ1 EHp-val.

S^(3,1)_t −0.051 −0.220 0.000 S^(6,3)_t −0.251 −0.153 0.000 S^(6,1)_t −0.060 −0.087 0.000

Notes: a) we have also used a Newey-West correction with a bandwidth ofm−1 but similar results were obtained with a fixed bandwidth of 12; b) when r^(n−m)_t+m is not available we have followed Hardouvelis (1994), usingr_t+m⁽ⁿ⁾ as a proxy.

on the other hand, the “puzzle” well known in the literature is also observed for the Portuguese case and our data clearly fail to pass the tests on the predictive ability of the spread. Bearing in mind that the latter conditions are the ones which better characterize the EH and that the former are insufficient to discriminate against other hypotheses of the term structure, we may conclude that it appears to be valid only in some weak, “asymptotic” or long-run form ³.

4.2 In multiple equation models

Concerning the multiple equation approach, Johansen’s ML method was imple- mented using PcGIVE 10.1 (Doornik and Hendry, 2001) and JMulTi 4.22 (L¨utkepohl and Kr¨atzig, 2004). Results for systems with 2 and 3 IMM interest rates are pre- sented below and, as previously mentioned, these were obtained including an un- restricted constant. However, all the procedures were also performed considering a restricted intercept, producing evidence which broadly agrees with the one which is presented.

In the modelling exercise we have faced two main problems: strong evidence for non-normality and for serial correlation of the disturbance vector. While non- Gaussianity is of no great concern (see, e.g., Gonzalo, 1994, and L¨utkepohl, 2004), the latter problem may impart somewhat fragile estimates and inferences. Obviously, augmenting the information set to cope with it is not an available option in the current context. Instead, we employed a robustifying strategy, considering several

3We borrow this term from the rational expectations hypothesis literature; see, e.g., Stein (1981) and Patterson (1987).

Table 5A. P-values of trace tests for cointegration: fixed lag lengths

p= 6 p= 12 p= 18

rates H0 λtrace λ^∗_trace λtrace λ^∗_trace λtrace λ^∗_trace r⁽¹⁾_t , r⁽³⁾_t r=0 0.001 0.003 0.004 0.014 0.000 0.004 r=1 0.542 0.556 0.165 0.198 0.122 0.171 r⁽¹⁾_t , r⁽⁶⁾_t r=0 0.024 0.037 0.003 0.012 0.001 0.010 r=1 0.545 0.559 0.198 0.233 0.172 0.227 r⁽³⁾_t , r⁽⁶⁾_t r=0 0.120 0.158 0.019 0.050 0.055 0.165 r=1 0.598 0.611 0.322 0.358 0.157 0.210 r⁽¹⁾_t , r⁽³⁾_t , r⁽⁶⁾_t r=0 0.007 0.021 0.012 0.096 – –

r=1 0.193 0.270 0.046 0.141 – –

r=2 0.530 0.552 0.234 0.290 – –

dynamic specifications.

Basically, we obtained results for two rather different types of dynamic specifi- cations, i. e., for fixed and for data dependent lag lengths (p). For the former, we used p = 6 and 12 for all systems and p = 18 only for bivariate systems. For the latter, besides resorting to the usual AIC and SC criteria, we have also employed a sequential general-to-specific (GS) strategy of eliminating insignificant lags based on likelihood ratio (LR) test statistics. When using the information criteria, we set p_max = 18 for bivariate systems and p_max = 12 for the trivariate case. For the GS-LR strategy, we used p_max= 12 and 6, respectively, and besides individual lag testing we have also used a joint confirmation test, testing all the restrictions imposed on the initial model.

Although maximum eigenvalue statistics were also computed for cointegration testing, we report only the evidence based on trace test statistics, which are more robust to non-Gaussianity. Besides the asymptotic p-values (denoted with λ_trace), tables 5A and 5B report also theirfinite sample corrected versions (λ^∗_trace).

Considering bivariate systems, previous evidence for cointegration is generally confirmed but it appears weaker for the two longer-term rates. Strong evidence for cointegration is found in the trivariate system but, more importantly, there is only very weak support that the cointegration rank is equal to two. Actually, this condition seems to hold only when the SC criterion for lag selection is used.

Table 5B. P-values of trace tests for cointegration: estimated lag lengths b

p_AIC pb_SC pb_LR

rates H0 pb λtrace λ^∗_trace pb λtrace λ^∗_trace pb λtrace λ^∗_trace r_t⁽¹⁾, r_t⁽³⁾ r=0 14 0.000 0.003 4 0.003 0.005 10 0.000 0.001

r=1 0.125 0.161 0.613 0.621 0.247 0.277

r_t⁽¹⁾, r_t⁽⁶⁾ r=0 18 0.001 0.010 1 0.011 0.013 11 0.005 0.016

r=1 0.172 0.227 0.664 0.667 0.214 0.245

r_t⁽³⁾, r_t⁽⁶⁾ r=0 18 0.055 0.165 1 0.000 0.000 11 0.023 0.052

r=1 0.157 0.210 0.693 0.696 0.295 0.328

r_t⁽¹⁾, r_t⁽³⁾, r⁽⁶⁾_t r=0 11 0.004 0.040 1 0.000 0.000 6 0.007 0.021

r=1 0.084 0.197 0.000 0.000 0.193 0.270

r=2 0.186 0.234 0.785 0.787 0.530 0.552

However, as is usually the case with SC, the chosen specification appears to be under-parameterized. As is well known, this tends to produce spurious finding for cointegration and for the number of cointegration vectors, and hence we give less weight to this evidence.

Taking these results into consideration, zero-sum restrictions regarding cointe- gration vectors were tested only in bivariate systems (see table 6). Now the evidence clearly tends to support the EH, confirming the one obtained with DOLS. Cointe- grating vector estimates vary between [1−0.95]⁰ and [1−0.99]⁰.

Table 6. P-values for cointegrating vector restriction tests

rates p= 6 p= 12 p= 18 bpAIC pbSC pbLR

r_t⁽¹⁾, r⁽³⁾_t 0.126 0.145 0.001 0.047 0.160 0.076 r_t⁽¹⁾, r⁽⁶⁾_t 0.150 0.213 0.025 0.025 0.074 0.182 r_t⁽³⁾, r⁽⁶⁾_t 0.194 0.458 0.265 0.265 0.015 0.263

Proceeding on the path of refining the restrictions required by the EH, table 7 contains the factor loading estimates (i.e., the αb_ij) and the p-values for weak exo- geneity tests. The empirical evidence supports theory: at the usual 5% significance level and with one exception only (in a case where SC is used), longer-term in- terest rates appear as weakly exogenous for the cointegration vectors. Moreover, confirming the evidence provided by the single equation approach, in every case the

estimates present the required sign, i. e., the spread predicts short rate changes in the expected direction.

Table 7. Factor loading estimates and p-values for weak exogeneity tests

p= 6 p= 12 p= 18 pˆ_AIC pˆ_SC pˆ_LR

vect. αb p-val. αb p-val. αb p-val. αb p-val. αb p-val. αb p-val.

r_t⁽¹⁾ -0.51 0.000 -0.60 0.000 -0.90 0.000 -0.75 0.000 -0.44 0.000 -0.57 0.000 r_t⁽³⁾ -0.19 0.058 -0.17 0.218 -0.22 0.223 -0.22 0.114 -0.17 0.044 -0.10 0.419 r_t⁽¹⁾ -0.33 0.000 -0.44 0.000 -0.62 0.000 -0.62 0.000 -0.27 0.000 -0.43 0.000 r_t⁽⁶⁾ -0.07 0.204 -0.04 0.576 -0.07 0.447 -0.07 0.447 0.02 0.724 -0.06 0.329 r_t⁽³⁾ -0.61 0.001 -0.54 0.010 -0.65 0.026 -0.65 0.026 -0.62 0.000 -0.61 0.003 r_t⁽⁶⁾ -0.27 0.068 -0.04 0.825 -0.09 0.698 -0.09 0.698 0.13 0.104 -0.14 0.423

5 Sample-split analysis

Although our dataset covers a relatively short span of time – less than 16 years –, with no single sharp and abrupt change in monetary and financial conditions, it is possible to distinguish between two sub-periods, according to the degree of stability and deregulation in those markets. The first sub-period, roughly corresponding to thefirst third of the sample, is characterized by some instability, high interest rates and a rather volatile behaviour of the spreads. Much of this instability is explained by some external shocks, related to events in European foreign exchange markets occurring at the initial stages of the European Monetary System (EMS)⁴. After some deregulation in the monetary andfinancial markets and in the context of a smoother EMS, in the middle of 1994⁵ interest rates began declining and a much more stable period initiated, both the spreads and the variation in interest rates exhibiting much less volatility (seefigure 1). Consequently, based on the historical description of the

4For a brief history of the Portuguese experience in the EMS see (in portuguese):

http://pt.wikipedia.org/wiki/Sistema Monet´ario Europeu. Camarero and Tamarit (2002)find some evidence of a regime shift in the cointegration relation between long and short interest rates for the Spanish case occurring in 1994. They attribute thisfinding to reasons similar to ours.

5“In the first half of 1994, ..., faced with the emergence of downward pressure on the escudo, the Banco de Portugal intervened in the foreign exchange market, while at the same time acting in the money market to effect a significant hike in its intervention rates... The return to normal exchange conditions,..., allowed the Banco de Portugal to reintroduce its market intervention rates at the beginning of July...”, Banco de Portugal (1995, pp. 39 and 44).

1990 1995 2000

−4

−3

−2

−1 0 1 2 3

4 S31 S61

Figure 1: The spreads S_t^(3,1) (S31) and S_t^(6,1) (S61)

facts and on a simple graphical analysis, we decided to split the sample in two: the first sub-period ends in June 1994 and the second one, containing almost ten years of data, begins in July 1994 (which represents the hypothesized break date).

A preliminary analysis provided strong support for our partition of the sample.

In a first step, we have analysed the stability of the VECM models derived from the VAR models of table 5B, imposing r = 1 but no further restrictions. Using the assumed break date, both break-point (CH_BP) and sample-split (CH_SS) Chow statistics tend to reject the stability hypothesis, with bootstrap p-values of 0.0000.

The only exceptions concern theCH_SS statistics for the models chosen using SC and for two of the models selected using LR tests (those for (r_t⁽¹⁾, r_t⁽³⁾) and (r_t⁽¹⁾, r_t⁽⁶⁾)).

In a second step, we have searched for the data point where the Chow statistics are maximized. With the exception of the dates estimated using CH_SS for the models chosen with SC, which appear a bit earlier, all the remaining estimated break dates are located around 1994:7, ranging between 1992:11 and 1995:5.

Examining whether the cointegration relationships might have suffered some regime shift provided further partial evidence for our hypothesis about the sample- split point. Towards this purpose we have used the supτ (fluctuation) test of Hansen

Figure 2: Fluctuation statistic for the largest eigenvalue of the VAR model for (r_t⁽³⁾, r_t⁽⁶⁾) with ˆp= 11.

and Johansen (1999), testing the constancy of the largest eigenvalue and concen- trating out the short-run parameters of the VECMs. In figures 2 and 3, where the dashed horizontal line represents the 5% critical value (1.36), we present two typical examples of this analysis: although the null hypothesis of constancy of the largest eigenvalue is not rejected at the 5% level, the time-path of the statistics exhibits some instability at the beginning of the sample and a more stable behaviour is observed only after the middle of 1994. The recursive estimates of the largest eigenvalues of the VAR models also show a rather similar behaviour: a common volatile pattern in the beginning of the sample which vanishes around 1994:7.

On the other hand, when the short-run dynamics is not concentrated out, i.e., when all the parameters are estimated recursively, a tendency for an increase in the evidence for instability is observed and in some cases the null of stability is rejected (actually, this occurs only for the longer models chosen with AIC). Hence, in general terms and contrasting with the finding of Camarero and Tamarit (2002) for the Spanish case, wefind much less evidence for a regime shift in the cointegration relations. Instead, it appears that the instability detected by the Chow statistics may be attributed mostly to the short-run dynamics.

Figure 3: Fluctuation statistic for the largest eigenvalue of the VAR model for (r_t⁽¹⁾, r_t⁽³⁾, r_t⁽⁶⁾) with ˆp= 6.

Subsequently we have proceeded considering July 1994 as the sample-split point and we have returned to the single equation approach analysis. In table 8 we present the results for DOLS estimation and testing that the cointegration parameter is unity, i.e., H₀ :β₁ = 1 vs. H₁ :β₁ 6= 1 inr⁽ⁿ⁾_t =β₀+β₁r^(m)_t +u_t. Although these results should be viewed with great caution since the first subsample is very short, the results for the whole sample period appear to be masking somewhat distinct situations: only after 1994:6 did the two longer-term rates adjust more closely to the behaviour of the shorter-term rate.

Table 8. DOLS estimates and tests

1989:1 — 2004:4 1989:1 — 1994:6 1994:7 — 2004:4 rates βˆ₀ βˆ₁ EH p-val. βˆ₀ βˆ₁ EHp-val. βˆ₀ βˆ₁ EHp-val.

r_t⁽³⁾, r_t⁽¹⁾ 0.045 1.032 0.129 3.968 0.785 0.000 -0.130 1.041 0.190 r_t⁽⁶⁾, r_t⁽³⁾ 0.080 1.009 0.551 0.111 1.010 0.755 -0.079 1.029 0.744 r_t⁽⁶⁾, r_t⁽¹⁾ 0.070 1.046 0.236 5.475 0.706 0.001 -0.073 1.043 0.733

In tables 9 and 10 we revisit the predictive implications of the EH, now using a sample-split perspective. The testing results confirm and reinforce the previous

evidence for the whole sample. In what concerns forecasting short rate changes (table 9), the forecasts for both sub-periods are in the correct direction but the EH is stillfirmly rejected. For the second sub-period this is somewhat surprising because a much more quiescent monetary and financial environment emerged in the second half of the nineties.

Table 9. The spread as predictor of short rate changes (equation (3))

1989:1 — 1994:6 1994:7 — 2004:4 spread ˆδ₀ ˆδ₁ EHp-val. δˆ₀ ˆδ₁ EH p-val.

S_t^(3,1) −0.051 0.228 0.000 −0.092 0.289 0.000

S_t^(6,3) −0.099 0.480 0.001 −0.133 0.152 0.000

S_t^(6,1) −0.254 0.532 0.000 −0.217 0.251 0.000

Note: for the calculation of the Wald test statistics we have used a Newey-West correction with a Bartlett kernel and a bandwidth of n−m, but similar results arise when a (fixed) bandwidth equal to 12 is employed.

On the other hand, we still find again strong evidence against the EH when the predictive ability of longer rate changes is considered (table 10). Although we observe that the coefficient ofS_t^(6,1) now appears correctly signed in the second sub- period, it is far from statistically significant because its p-value is equal to 0.786.

Table 10. The spread as a predictor of long rate changes (equation (6)) 1989:1 — 1994:6 1994:7 — 2004:4

ˆλ₀ λˆ₁ EHp-val. λˆ₀ λˆ₁ EHp-val.

S_t^(3,1) 0.020 −0.245 0.000 −0.090 −0.178 0.000

S_t^(6,3) −0.198 −0.039 0.000 −0.265 −0.695 0.001

S_t^(6,1) 0.098 −0.160 0.000 −0.098 0.110 0.027

Notes: a) we have also used a Newey-West correction with a bandwidth ofm−1 but similar results were obtained with afixed bandwidth of 12; b) whenr^(n−m)_t+m is not available we have followed Hardouvelis (1994), usingrt+m⁽ⁿ⁾ as a proxy.

6 Concluding Remarks

As far as we know, this is the first time that Portuguese IMM rates are used to test the EH. In general terms, mixed but very weak evidence is provided by these

data. Notwithstanding the mostly unfavourable evidence for the requirement that the cointegration rank equals two in trivariate systems, strong support is found only when general, long-run implications, are under scrutiny. When more detailed and demanding conditions are tested, the supporting evidence either becomes much weaker or vanishes completely.

In particular, the EH “puzzle” is also observed for the Portuguese case: although its forecasts of future short-term rates are in the correct direction, the spread between longer and shorter rates provides a forecast in the wrong direction for the behaviour of longer rates. Moreover, all the test results concerning the predictive ability of the spread are totally at odds with the EH. Since our dataset covers only the short end of the maturity spectrum, thesefindings are consistent with most empirical evidence for other countries.

Rather than viewing these results as “paradoxical”, we follow Thornton (2006) and interpret them as invalidating the core of the EH. Hence, only a very weak,

“asymptotic” version of the hypothesis, appears to hold for the Portuguese case.

Afinal remark refers to the robustness of these results to the sample period under scrutiny. The sample-split analysis allowed us to confirm and to strengthen them.

All the inference procedures for the single equation setup were replicated for the two sub-samples but we could not find any sharply contrasting difference between the results. A slight increase in the evidence favouring the EH is observed in the calmer second sub-period but this concerns only the long-run properties of the data and may be attributed to the poor performance of the methods used on the (very) small sample of the first sub-period. In other words, our evidence concerning the EH appears to be robust to the sample period.

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